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MA6151_MATHEMATICS-I_QBank_ACY 2016-17(ODD) COMMON TO ALL BRANCHES OF I SEMESTER B.E., B.TECH. PREPARED BY DEPARTMENT OF MATHEMATICS 1

QUESTION BANK

SUBJECT : MA6151 – MATHEMATICS -1

SEM / YEAR:I Sem / I year B.E.B.Tech. (Common to all branches)

UNIT I - MATRICES

Eigen values and Eigen vectors of a real matrix - Characteristic equation - Properties of eigen values and

eigenvectors - Statement and applications of Cayley-Hamilton Theorem - Diagonalization of matrices -

Reduction of a quadratic form to canonical form by orthogonal transformation - Nature of quadratic forms.

Q.No. Question

Bloom’s Taxonomy

Level

Domain

PART – A

1. Find the sum and product of all the Eigen values of ( −− −− )

BTL -2 Understanding

2. What are the Eigen values of the matrix A + 3I, if the Eigen

values of the matrix � = −− are 6 and -1? Why? BTL -1 Remembering

3. Find the Eigen vales of 3A + 2I, where � = BTL -2 Understanding

4. Find the Eigen values of the inverse of the matrix

� = ( ) BTL -2 Understanding

5. If λ is the Eigen value of the (a square) matrix A, then prove that λ2

is the Eigen value of A2

BTL -3 Applying

6. If the Eigen values of the matrix A of order 3 x 3 are 2, 3 and 1,

then find the Eigen values of adjoint of A. BTL -2 Understanding

7. Find the values of a and b such that the matrix has 3 and

-2 its Eigen values


8. The product of two eigen values of the matrix � = ( −− −− ) is 16. Find the third eigen value of A.


9. If 3 and 6 are two Eigen values of � = ( ), write down

all the Eigen values of A-1

BTL -3 Applying

10. State Cayley – Hamilton theorem. BTL -1 Remembering

11. Find the Eigen values of � = ( − − ). Also find the Eigen

values of -3A


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12. Find the symmetric matrix A, whose eigen values are 1 and 3

with corresponding eigenvectors (− ) ( ) BTL -2 Understanding

13. Find the Eigen values of A

-1 where� = ( ) BTL -2 Understanding

14. If 2, -1, -3 are the Eigen values of the matrix A, then find the

Eigen values of the matrix A2 – 2I


15. Can � = be diagonalised? Why? BTL -4 Analyzing

16. What is the nature of the quadratic form x2 + y

2 + z

2 in four

variables? BTL -2 Understanding

17. Identify the nature, index and signature of the quadratic form + + BTL -4 Analyzing

18. Give the nature of the quadratic form whose matrix is (− − − )

BTL -4 Analyzing

19. Write down the matrix of the quadratic form + + + − BTL -3 Applying

20. Write down the quadratic form corresponding to the matrix � = ( −− )

BTL -3 Applying

PART – B

1.(a) Find the Eigen values and Eigen vectors of ( ) BTL -1

Remembering

1. (b) If A = ( −− − − ), verify Cayley – Hamilton Theorem and

hence find A-1

BTL -4 Applying

2. (a)

Find the Eigen values and Eigen vectors of a matrix

� = (− −− − ) BTL -2 Understanding

2.(b) Reduce the quadratic form x2+5y

2+z

2+2xy+2yz+6zx into

canonical form and hence find its rank. BTL -3 Applying

3. (a)

State Cayley – Hamilton theorem and using it, find the matrix

represented by A8-5A

7+7A

6-3A

5+A

4-5A

3+8A

2-2A+I when � = ( )


3.(b) Reduce the quadratic form + + − −+ into canonical form by the orthogonal

transformation

BTL -3 Applying

4. (a)

If �� for (i = 1, 2, …, n) are the non-zero Eigen values of A ,

then prove that (1) �� are the Eigen Values of ��, where K

being a non-zero scalar; (2) ��are the Eigen values of �− ,

(3) �� are the Eigen Values of ��

BTL -4

Analyzing

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4.(b) Reduce the quadratic form 3x

2+ 5y

2+ 3z

2- 2yz + 2zx - 2xy into

canonical form through orthogonal transformation.

BTL -3 Applying

5. (a)

Using Cayley-Hamilton theorem find A-1

and A4, if � = ( −− − )

BTL -1 Remembering

5.(b) Determine a diagonal matrix orthogonally similar to the real

symmetric matrix( −− ) BTL -2 Understanding

6. (a)

Verify Cayley – Hamilton theorem for the matrix � = .

Also compute A-1

BTL -4 Analyzing

6.(b)

Reduce the quadratic form + − into the

canonical form by an orthogonal reduction. Also find its nature BTL -3 Applying

7. (a)

Diagonalize the matrix � = ( ) BTL -4 Analyzing

7. (b)

Verify Cayley-Hamilton theorem for the matrix � = ( − −− ) BTL -4 Analyzing

8. (a)

Show that the matrix( − )satisfies its own characteristic

equation. Find also its inverse

BTL -4 Analyzing

8.(b) Reduce the quadratic form + + into

canonical form BTL -3 Applying

9. (a)

The eigenvectors of a 3x3 real symmetric matrix A

corresponding to the eigen values 2,3, 6 are (1, 0, -1)T,

(1, 1, 1)T

and (1, 2, -1)T

respectively. Find the matrix A. BTL -2 Understanding

9.(b) Show that the matrix � = ( −− −− )satisfies its own

characteristic equation. Find also its inverse.

BTL -4 Analyzing

10.(a)

If the eigen values of � = ( −− −− )are 0, 3, 15, find the

eigen vectors


10.(b) Find An using Cayley-Hamilton theorem, taking� = [ ].

Hence find A3


11.(a) Find the Eigen vales and Eigen vectors of the matrix ( ) BTL -2 Understanding

11.(b)

Determine a diagonal matrix orthogonally similar to the real

symmetric matrix ( −− −− ) BTL -2 Understanding

12.(a) Using Cayley – Hamilton theorem find A

4 for the matrix � = ( −− )


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12.(b) Determine a diagonal matrix orthogonally similar to the real

symmetric matrix (− −−− − ). BTL -2 Understanding

13.(a) Using Cayley Hamilton theorem find the inverse of BTL -2 Understanding

13.(b) Reduce the quadratic form x

2+ y

2+ z

2- 2xy - 2yz - 2zx into

canonical form through an orthogonal transformation. Write

down the transformation.

BTL -3 Applying

14.(a) Using Cayley Hamilton theorem for the matrix find the

value of the polynomial � − � − � + � − � − � BTL -2 Understanding

14.(b) Reduce the Quadratic form x + y + z + xy to canonical

form by orthogonal reduction and states its nature. BTL -3 Applying

UNIT -II SEQUENCE AND SERIES

Sequences: Definition and examples – Series: Types and Convergence – Series of positive terms – Tests of

convergence: Comparison test, Integral test and D’Alembert’s ratio test – Alternating series – Leibnitz’s test –

Series of positive and negative terms – Absolute and conditional convergence.

Q.No. Question Bloom’s

Taxonomy

Level

Domain

PART - A

1. Distinguish between a sequence and series. BTL -1 Remembering

2. Discuss the convergence of the sequence { } where = +. BTL -1 Remembering

3. Discuss the convergence of the sequence {� } where � = −+ . BTL -1 Remembering

4. Examine the convergence of the series ∑ log +∞= . BTL -1 Remembering

5. Test the convergence of the series + + + + ⋯ BTL -3 Applying

6. Using Comparison test, prove that the series ⋅ + ⋅ + ⋅ + ⋯

is divergent. BTL -3 Applying

7. Find the nature of the series + + + ⋯ . BTL -2 Understanding

8. Test the convergence of the series + √ + √ + √ + ⋯ + √ +. .. BTL -3 Applying

9. Test the convergence of the series ∑ +∞= . BTL -4 Analyzing

10. Test the convergence of the series ∙ ! + ∙ ! + ∙ ! + ⋯ BTL -3 Applying

11. Define integral test. BTL -1 Remembering

12. Using integral test determine the convergence of the series + + + ⋯ + − + ⋯.


13. Test the convergence of the series + − − + + −− + ⋯

BTL -3 Applying

14. Examine the convergence of the series − + − + ⋯ BTL -2 Understanding

15. Test the convergence of the series � � + � � + � � + ⋯ BTL -4 Analyzing

16. Test the convergence of the series ∑ −∞= BTL -4 Analyzing

17. Test the convergence of the series ∑ −∞= BTL -4 Analyzing

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18. Test the convergence of the series ∑ − + √∞= . BTL -4 Analyzing

19. Test the convergence of the series ∑ − �−−∞= . BTL -4 Analyzing

20. Give an example for conditionally convergent series. BTL -6 Creating

PART -B

1.(a) Show by direct summation of n terms that the series ⋅ + ⋅ +

⋅ + ⋯ is convergent. BTL -3 Applying

1. (b) Test the convergence of the series + + ∙ ∙ + ∙ ∙ ∙ ∙ +⋯

BTL -4 Analyzing

2. (a)

Using Comparison test, examine the convergence or divergence

of ∙ ∙ + ∙ ∙ + ∙ ∙ + ⋯ BTL -2 Understanding

2.(b) Examine the convergence and the divergence of the series + + + + ⋯ + �−�+ − + ⋯ > .

BTL -2

Understanding

3.(a) Test the convergence of the series ∑ √ + − ∞= BTL -4 Analyzing

3.(b) Discuss the convergence of the series ∑ √√ + BTL -2 Understanding

4. (a) Test the convergence of the series ∙ ∙ + ∙ ∙ + ∙ ∙ +⋯

BTL -3 Applying

4.(b) Test the convergence of the series ∑ +∞= , > BTL -4 Analyzing

5. (a) Examine convergence of the series ∑ √ + − ∞= BTL -3 Applying

5.(b) Test the convergence of the series + �! + �! + �! + ⋯ by

D’Alembert’s ratio test BTL -4 Analyzing

6. (a) Examine the convergence of the series +√ + √ +√ + √ +√ + ⋯ BTL -4 Analyzing

6.(b)

Test the convergence of the series

+ . . + . . . . + ⋯by Ratio test BTL -4 Analyzing

7. (a)

Using D’ Alembert ratio test, examine the convergence or divergence of + + + ⋯

BTL -4 Analyzing

7. (b) Test for absolute convergence of the series + ! + ! + ! + ⋯ BTL -2 Understanding

8. (a) Prove that the harmonic series is divergent BTL -1 Remembering

8.(b) Prove that the series + . + . + . + ⋯ is divergent by

Ratio test BTL -5 Evaluating

9. (a) Test the convergence of the series ∑ +∞= by Integral test BTL -3 Applying

9.(b) Determine convergence of an alternating series ∑ c s �+∞= and

also test for absolute and conditional convergence. BTL -3 Applying

10.(a) Test the convergence of the series ∑ √l g�∞= BTL -3 Applying

10.(b) Find integral of the convergence + − + + + − + … < <


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11.(a) Test the convergence of the series ∑ l g∞= BTL -3 Applying

11.(b) Find the convergence of the series − √ + √ − √ + ⋯∞ BTL -2 Understanding

12.(a) Find the nature of the series ∑ l g �∞= by Cauchy’s integral test.

BTL -5 Evaluating

12.(b) Test the convergence and divergence of the series √ + − √ + +√ + − √ + + ⋯

BTL -4 Analyzing

13.(a) Test the convergence of the series ∑ −∞= BTL -4 Analyzing

13.(b) Test the convergence or divergence of ⋅ − ⋅ + ⋅ − ⋅ + ⋯ BTL -3 Applying

14.(a) Discuss the convergence of ∑ sin∞= BTL -4 Analyzing

14.(b) Test the convergence of the series − + ++ + − + + + + ⋯

BTL -3

Applying

UNIT – III APPLICATION OF DIFFERENTIAL CALCULUS

Curvature in Cartesian co-ordinates – Centre and radius of curvature – Circle of curvature – Evolutes –

Envelopes - Evolute as envelope of normals.


Taxonomy

Level

Domain

PART - A

1. What is Circle of Curvature? BTL -1 Remembering

2. Find the curvature of the curve of the curve

+ + − + = . BTL -2 Understanding

3. Find the radius of curvature of the curve = � at (c, c).


4. Find radius of curvature of curve + − + − = . BTL -2 Understanding

5. What is the radius of curvature of the curve + = at the

point (1,1) BTL -1 Remembering

6. Find the radius of curvature for = at the point where it is cuts

the y-axis. BTL -2 Understanding

7. Find the curvature of the curve + + + − = at , . BTL -2 Understanding

8. What is the radius of curvature of the curve + = at the

point , . BTL -1 Remembering

9. Define curvature of a Plane curve. BTL -1 Remembering

10. What is the curvature of the circle − + + = at

any point on it . BTL -3 Applying

11. Find the envelope of cos � + sin � = , where � is a

parameter. BTL -2 Understanding

12. Find the envelope of the family of circles − � + = �,

where � is the parameter BTL -2 Understanding

13. Find the envelope of the lines ,1sincos

b

y

a

xbeing the

parameter.


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14. Find the envelope of the lines cosec � − cot � = , � being

the parameter. BTL -2 Understanding

15. Find the envelope of the family of lines tcyt

t

x,2 being a

parameter.


16. Write the properties of evolutes. BTL -1 Remembering

17. Find the envelope of − sin � = cos �, where � is a


18. Find the envelope of the family of straight lines .

m

amxy

where a is a parameter.


19. Find the envelope of the family of straight lines

= − − where is a parameter. BTL -2 Understanding

20. Find the Envelope of .222 bmamxy where is a


PART -B

1.(a) Find radius of the curvature of .32

32

32

ayx BTL -2

Understanding

1. (b) Find the centre of curvature of xyyx 633 at (3, 3).


2. (a)

Show that the radius of curvature at any point of the Catenary

c

xhCy cos is C. Also find at (0, c). BTL -3 Applying

2.(b) Prove that for the curve22

3

2

2,

x

y

y

x

axa

axy

. BTL -3 Applying

3. (a) Find the radius of curvature at any point of the Cycloid

.cos1,sin ayax BTL -2 Understanding

3.(b) Find the circle of curvature at

4,

4

aa on ayx . BTL -2 Understanding

4. (a) Find the radius of curvature and centre of curvature of the

Parabola .42 axy at the point t. BTL -2 Understanding

4.(b)

If the centre of curvature of an ellipse ,12

2

2

2

b

y

a

x at one end on

the minor axis lie at the other end, Prove that the eccentricity of

the ellipse is .2

1

BTL -3 Applying

5. (a)

Find the radius of curvature of the point

2

3,

2

3 aa on the curve

.333 axyyx


5.(b) Find the equation of the circle of curvature of the rectangular

hyperbola 12xy at (3, 4). BTL -2

Understanding

6. (a) Obtain the evolute of cos1,sin ayax . BTL -6 Creating

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6.(b) Obtain the envelope ,1b

y

a

x if 2222 , cabiicbai . BTL -6 Creating

7. (a)

Find the equation of the evolute of the parabola ,42 ayx treating it as the envelope of normal.


7. (b)

Find the envelope of the ellipse ,12

2

2

2

b

y

a

x where a and b are

connected by the relation nnn cba , c being a constant.


8. (a) Considering the evolute as the envelope of normal, Find the

evolute of the tractrix .sin,2

tanlogcos ayax

BTL -2

Understanding

8.(b)

Prove that the radius of curvature at any point (x, y) on

.)(2

12

3

2

1

2

1

ab

byaxis

b

y

a

x

BTL -3 Applying

9. (a) Show that the radius of curvature at any point of the curve

cossin,cossin aeyaex is twice the

perpendicular distance from the origin to the tangent at the point.

BTL -3 Applying

9.(b) Find the equation of the evolute of the ellipse .12

2

2

2

b

y

a

x BTL -2 Understanding

10.(a) Find the evolute of the rectangular hyperbola .2Cxy BTL -2 Understanding

10.(b) Find the points on the parabola .42 xy at which the radius of

curvature is .24 BTL -2

Understanding

11.(a) Find the equation of the evolute of the parabola .42 axy BTL -2 Understanding

11.(b) Considering the evolute as the envelope of the normal, find the

evolute of the asteroid .32

32

32

ayx BTL -2

Understanding

12.(a) Find the envelope of the ellipse ,12

2

2

2

b

y

a

x where a and b are

connected by the relation 222 cba , c being a constant.


12.(b) Find the equation of the evolute of the hyperbola .12

2

2

2

b

y

a


13.(a) Find the envelope of ,1

m

y

l

x where l & m are connected by

the relation ,1b

m

a

l where and are constants. BTL -2

Understanding

13.(b) Show that the evolute of the cycloid

cos1,sin ayax is another cycloid. BTL -3 Applying

14.(a) Find the envelope of the family of straight lines

,cossinsincos cyx being the parameter. BTL -2

Understanding

14.(b) Find the envelope of the straight line ,1

b

y

a

x where the

parameter and are connected by the relation ccba nnn ,being a constant.


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UNIT – IV DIFFERENTIAL CALCULUS OF SEVERAL VARIABLES

Limits and Continuity – Partial derivatives – Total derivative – Differentiation of implicit functions –

Jacobian and properties – Taylor’s series for functions of two variables – Maxima and minima of functions of

two variables – Lagrange’s method of undetermined multipliers


Taxonomy

Level

Domain

PART - A

1. Evaluate .2

5lim

22

2yx

xy

yx

BTL -5 Evaluating

2. If .,z

uz

y

uy

x

uxfindthen

y

x

x

z

z

yu


3. If .,,,z

u

y

u

x

ufindthenxzzyyxfu


4. If + = , then finddx

dy. BTL -2 Understanding

5. Find the value of dt

du, given atyatxaxyu 4,,4 22 BTL -2 Understanding

6. If .,223223

dt

dufindthenatyandatxwhereyxyxu BTL -2 Understanding

7. Find .,,sin 2tyexwherey

xuif

dt

du t


8. Find .log,, tyexwherey

xuif

dt

du t BTL -2 Understanding

9. Find the Jacobian .sin&cos,

,

, ryrxif

yx

r


10.

Find the Jacobian

22,2,sin&cos,,

,yxvxyuryrxif

r

vu

, without

actual substitution.


11. If ,22

222

x

yxvand

x

yu

.,

,

yx

vufind


12. If .

,

,.1,1

vu

yxFinduvyvux


13. If = show that xy

u

yx

u

22

BTL -3 Applying

14. If u = ),(

),(,tantan

1

11

yx

vufindyxvand

xy

yx

BTL -3 Applying

15. Find the Taylor series expansion of near the point , up to

first term

BTL -3 Applying

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16. Expand )2(&)1(232 yxofpowersinyxxy , using

Taylor’s theorem up to first degree form BTL -4 Analyzing

17. Find the Stationary points of

.7215153, 2223 xyxxyxyxf BTL -2 Understanding

18. Find the Stationary points of .222 yxyxyx BTL -2 Understanding

19. State the Sufficient condition for yxf , to be extremum at a point BTL -1 Remembering

20. Find the minimum point of f(x,y) = 12622 xyx . BTL -2 Understanding

PART -B

1.(a) If u = log (x2 + y

2 ) +

x

y1tan , prove that + = BTL -3

Applying

1. (b) If findz

xywand

y

zxv

x

zyu ,,

.

,,

,,

zyx

wvu


2. (a) If ,tantan 1212

y

xy

x

yxu then prove that

22

222

yx

yx

yx

u

BTL -3 Applying

2.(b) Find the Jacobian of

� , ,� ,�,� of the transformation

= � � �, = � � � �, = � BTL -2 Understanding

3. (a) If z is a function of and and = + − , = − − then

show that �� − �� = �� − ��

BTL -3 Applying

3.(b)

Find the shortest & longest distance from the point(1, 2,-1) to the

sphere .24222 zyx Using Lagrange’s multiplier method of constrained Maxima and Minima


4. (a) ,,, uvwzuvzyuzyxIf vuwvu

zyxthatprove 2

),,(

),,(

BTL -3 Applying

4.(b)

sin,cos),( ryrxwhereyxfuIf

2

2

2221

u

rr

u

y

u

x

uthatprove BTL -4 Analyzing

5. (a) Transform equation 02 yyxyxx zzz by changing the

independent variables using .yxvandyxu BTL -4 Analyzing

5.(b)

Verify whether the following functions are functionally

dependent, and if so, find the relation between them = +− , = − + − BTL -3 Applying

6. (a)

uvyvuxwhereyxfzIf 2),( 22

2

2

2

222

2

2

2

2

)(4y

z

x

zvu

v

z

u

zthatprove BTL -3 Applying

6.(b) Expand

2,1sin

atxy up to second degree terms using Taylor’s

series. BTL -4 Analyzing

7. (a) Expand ye x 1log in powers of & up to terms of third

degree terms using Taylor’s series BTL -4 Analyzing

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7. (b) Discuss the maxima and minima of , = − − . BTL -1 Remembering

8. (a) Find the Taylors series expansion of � at the point (-1,

4

)

up to the third degree terms


8.(b) Find the extreme value of

222 zyx subject to the condition

.3azyx BTL -2 Understanding

9. (a) Expand x

y1tan in the neighborhood of (1, 1) BTL -4 Analyzing

9.(b) Find the Maximum value of .azyxwhenzyx pnm BTL -2 Understanding

10.(a) Find the Taylors series expansion of x

2y

2+2x

2y+3xy

2 in powers of + and − up to 3rd degree


10.(b) Find the maximum value of .,0sinsinsin yxwhereyxyx BTL -2 Understanding

11.(a) Expand in powers of − and − upto third degree

terms by Taylor’s series BTL -4 Analyzing

11.(b) Find the minimum value of .3232 azyxtosubjectyzx BTL -2 Understanding

12.(a) Expand Taylor’s series of 233 xyyx in powers of − and −

BTL -4 Analyzing

12.(b)

Find the volume of the greatest rectangular parallelepiped that can

be inscribed in the ellipsoid .12

2

2

2

2

2

c

z

b

y

a


13.(a) Find the extremum value of .20123, 33 yxyxyxf BTL -2 Understanding

13.(b)

If = + + , = + + = + + ,

determine whether there is a functional relationship between , ,

and if so, find it.

BTL -5 Evaluating

14.(a)

A rectangular box open at the top is to have volume of 32 cm. Find

the dimension of the box requiring least material for its

Construction BTL -6 Creating

14.(b) If = , = + + = + + then find .,,

,,

wvu

zyx


UNIT – V MULTIPLE INTEGRALS

Double integrals in Cartesian and polar coordinates – Change of order of integration – Area enclosed by

plane curves – Change of variables in double integrals – Area of a curved surface - Triple integrals – Volume

of Solids


Taxonomy

Level

Domain

PART - A

1. Evaluate BTL -5 Evaluating

2. Evaluate �� / BTL -5 Evaluating

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3. Find the area bounded by the lines = , = = BTL -2 Understanding

4. Evaluate �� BTL -5 Evaluating

5. Evaluate + BTL -5 Evaluating

6. Evaluate √ −

BTL -5 Evaluating


8. Evaluate � � �� BTL -5 Evaluating


10. Evaluate over the region bounded by = , = , = =

BTL -5 Evaluating

11. Change the order of integration , BTL -3 Applying

12. Change the order of integration , BTL -3 Applying

13. Change the order of integration , ∞∞ BTL -3 Applying

14. Evaluate + + over the region bounded by = , = , = = , = , =

BTL -5 Evaluating

15. Write down the double integral to find the area of the circles = � �, = � �

BTL -1 Remembering

16. Evaluate +√ BTL -5 Evaluating


18. Evaluate + + BTL -5 Evaluating


20. Evaluate + + BTL -5 Evaluating

PART-B

1.(a) Evaluate over the positive quadrant of the circle x2 +

y2 = a

2

BTL -5

Evaluating

1. (b) Change the order of integration −∞

and hence

evaluate it

BTL -3 Applying

2. (a) Evaluate + √ − by changing into polar

coordinates. BTL -5

Evaluating

2.(b) By change the order of integration and evaluate

− BTL -3

Applying

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3. (a) Evaluate over the positive quadrant of the circle + = BTL -5

Evaluating

3.(b) Change the order of integration −� and hence

evaluate it BTL -3

Applying

4. (a)

By changing in to polar Co – ordinates , evaluate ∫ ∫ − +∞∞

BTL -3

Applying

4.(b) Change the order of integration − and hence

evaluate it BTL -3

Applying

5. (a) Evaluate +√ − by changing into polar co –

ordinates BTL -5

Evaluating

5.(b) Change the order of integration √ + and hence

evaluate it BTL -3

Applying

6. (a) Change in to polar Co – ordinates and evaluate √ + BTL -3 Applying

6.(b) Find the area of the cardioids = + � BTL -2 Understanding

7. (a)

Change in to polar Co – ordinates and evaluate

√ + √ −

BTL -3 Applying

7. (b) Find the volume of the tetrahedron bounded by the coordinate

planes and + + = 1. BTL -2

Understanding

8. (a) Find the area enclosed by the curve = and the lines + = , = . BTL -2 Understanding

8.(b) Evaluate + + + where V is the region bounded by = , = , = and + + = .

BTL -5 Evaluating

9. (a) Find the area included between the curves = and

= BTL -2

Understanding

9.(b) Evaluate � BTL -5

Evaluating

10.(a) Find the smaller area bounded by = and + = . BTL -2

Understanding

10.(b) Find the volume of the ellipsoid + + = BTL -2 Understanding

11.(a) Find the area which is inside the circle = � and outside

the cardioids = + � . BTL -2 Understanding

11.(b) Evaluate √ − − − √ − − √ −

BTL -5 Evaluating

12.(a) Find the area that lies inside the cardioids = + � and

outside the circle = by double integral BTL -2

Understanding

12.(b) Find the volume of sphere + + = using triple

integral BTL -2

Understanding

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13.(a) Evaluate √ + � by converting into polar

coordinates where R is the first quadrant part of the region

bounded by two circles + = + =

BTL -5 Evaluating

13.(b) Find the volume bounded by the cylinder + = and the planes + + = , = BTL -2

Understanding

14.(a) Evaluate + + BTL -5 Evaluating

14.(b) Find the area enclosed by the curves = and = BTL -2

Understanding

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